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 ChebyshevT

 http://functions.wolfram.com/07.04.06.0074.01

 Input Form

 ChebyshevT[\[Nu], z] == Subscript[F, Infinity][z, \[Nu]] /; (Subscript[F, m][z, \[Nu]] == 2^(\[Nu] - 1) (z - 1)^\[Nu] Sum[((Pochhammer[-\[Nu], k] Pochhammer[1/2 - \[Nu], k])/ (Pochhammer[1 - 2 \[Nu], k] k!)) (2/(1 - z))^k, {k, 0, m}] + (2^(-\[Nu] - 1) Sum[((Pochhammer[\[Nu], k] Pochhammer[1/2 + \[Nu], k])/ (Pochhammer[1 + 2 \[Nu], k] k!)) (2/(1 - z))^k, {k, 0, m}])/ (z - 1)^\[Nu] == ChebyshevT[\[Nu], z] + ((2^(m + \[Nu]) (-1)^m Pochhammer[1/2 - \[Nu], 1 + m] Pochhammer[-\[Nu], 1 + m] (z - 1)^(\[Nu] - m - 1))/ ((1 + m)! Pochhammer[1 - 2 \[Nu], 1 + m])) HypergeometricPFQ[ {1, 1 + m - \[Nu], 3/2 + m - \[Nu]}, {2 + m, 2 + m - 2 \[Nu]}, 2/(1 - z)] + ((2^(m - \[Nu]) (-1)^m Pochhammer[\[Nu], 1 + m] Pochhammer[1/2 + \[Nu], 1 + m])/((1 + m)! Pochhammer[1 + 2 \[Nu], 1 + m])) (z - 1)^(-\[Nu] - m - 1) HypergeometricPFQ[ {1, 1 + m + \[Nu], 3/2 + m + \[Nu]}, {2 + m, 2 + m + 2 \[Nu]}, 2/(1 - z)] && Element[m, Integers] && m >= 0) && !Element[2 \[Nu], Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ChebyshevT", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", RowBox[List["z", ",", "\[Nu]"]], "]"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["F", "m"], "[", RowBox[List["z", ",", "\[Nu]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List["\[Nu]", "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "\[Nu]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", RowBox[List["2", " ", "\[Nu]"]]]], ",", "k"]], "]"]], RowBox[List["k", "!"]]]]], " ", SuperscriptBox[RowBox[List["(", FractionBox["2", RowBox[List["1", "-", "z"]]], ")"]], "k"]]]]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List["-", "\[Nu]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]], ",", "k"]], "]"]], RowBox[List["k", "!"]]]]], " ", SuperscriptBox[RowBox[List["(", FractionBox["2", RowBox[List["1", "-", "z"]]], ")"]], "k"]]]]]]]]], "\[Equal]", RowBox[List[RowBox[List["ChebyshevT", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["m", "+", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", RowBox[List["1", "+", "m"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["1", "+", "m"]]]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List["\[Nu]", "-", "m", "-", "1"]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "m"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", RowBox[List["2", " ", "\[Nu]"]]]], ",", RowBox[List["1", "+", "m"]]]], "]"]]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "+", "m", "-", "\[Nu]"]], ",", RowBox[List[FractionBox["3", "2"], "+", "m", "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "m"]], ",", RowBox[List["2", "+", "m", "-", RowBox[List["2", " ", "\[Nu]"]]]]]], "}"]], ",", FractionBox["2", RowBox[List["1", "-", "z"]]]]], "]"]]]], " ", "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["m", "-", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", RowBox[List["1", "+", "m"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List["1", "+", "m"]]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "m"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]], ",", RowBox[List["1", "+", "m"]]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "m", "-", "1"]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "+", "m", "+", "\[Nu]"]], ",", RowBox[List[FractionBox["3", "2"], "+", "m", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "m"]], ",", RowBox[List["2", "+", "m", "+", RowBox[List["2", " ", "\[Nu]"]]]]]], "}"]], ",", FractionBox["2", RowBox[List["1", "-", "z"]]]]], "]"]]]]]]]], " ", StyleBox[")", Rule[FontWeight, "Plain"]]]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]]]], ")"]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List[RowBox[List["2", "\[Nu]"]], ",", "Integers"]], "]"]], "]"]]]]]]]]

 MathML Form

 T ν ( z ) F ( z , ν ) /; ( ( F m ( z , ν ) 2 - ν - 1 ( z - 1 ) - ν k = 0 m ( ν ) k TagBox[SubscriptBox[RowBox[List["(", "\[Nu]", ")"]], "k"], Pochhammer] ( ν + 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], ")"]], "k"], Pochhammer] ( 2 ν + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], ")"]], "k"], Pochhammer] k ! ( 2 1 - z ) k + 2 ν - 1 ( z - 1 ) ν k = 0 m ( - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], "k"], Pochhammer] ( 1 2 - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ")"]], "k"], Pochhammer] ( 1 - 2 ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], "k"], Pochhammer] k ! ( 2 1 - z ) k 2 m - ν ( - 1 ) m ( ν ) m + 1 TagBox[SubscriptBox[RowBox[List["(", "\[Nu]", ")"]], RowBox[List["m", "+", "1"]]], Pochhammer] ( ν + 1 2 ) m + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], ")"]], RowBox[List["m", "+", "1"]]], Pochhammer] ( z - 1 ) - m - ν - 1 ( m + 1 ) ! ( 2 ν + 1 ) m + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], ")"]], RowBox[List["m", "+", "1"]]], Pochhammer] 3 F 2 ( 1 , m + ν + 1 , m + ν + 3 2 ; m + 2 , m + 2 ν + 2 ; 2 1 - z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "+", "\[Nu]", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "+", "\[Nu]", "+", FractionBox["3", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["m", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "+", RowBox[List["2", " ", "\[Nu]"]], "+", "2"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[FractionBox["2", RowBox[List["1", "-", "z"]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] + T ν ( z ) + 2 m + ν ( - 1 ) m ( 1 2 - ν ) m + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ")"]], RowBox[List["m", "+", "1"]]], Pochhammer] ( - ν ) m + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], RowBox[List["m", "+", "1"]]], Pochhammer] ( z - 1 ) - m + ν - 1 ( m + 1 ) ! ( 1 - 2 ν ) m + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], RowBox[List["m", "+", "1"]]], Pochhammer] 3 F 2 ( 1 , m - ν + 1 , m - ν + 3 2 ; m + 2 , m - 2 ν + 2 ; 2 1 - z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "-", "\[Nu]", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "-", "\[Nu]", "+", FractionBox["3", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["m", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "-", RowBox[List["2", " ", "\[Nu]"]], "+", "2"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[FractionBox["2", RowBox[List["1", "-", "z"]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] m ) ¬ 2 ν TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] ) Condition ChebyshevT ν z Subscript F z ν Subscript F m z ν 2 -1 ν -1 z -1 -1 ν k 0 m Pochhammer ν k Pochhammer ν 1 2 k Pochhammer 2 ν 1 k k -1 2 1 -1 z -1 k 2 ν -1 z -1 ν k 0 m Pochhammer -1 ν k Pochhammer 1 2 -1 ν k Pochhammer 1 -1 2 ν k k -1 2 1 -1 z -1 k 2 m -1 ν -1 m Pochhammer ν m 1 Pochhammer ν 1 2 m 1 z -1 -1 m -1 ν -1 m 1 Pochhammer 2 ν 1 m 1 -1 HypergeometricPFQ 1 m ν 1 m ν 3 2 m 2 m 2 ν 2 2 1 -1 z -1 ChebyshevT ν z 2 m ν -1 m Pochhammer 1 2 -1 ν m 1 Pochhammer -1 ν m 1 z -1 -1 m ν -1 m 1 Pochhammer 1 -1 2 ν m 1 -1 HypergeometricPFQ 1 m -1 ν 1 m -1 ν 3 2 m 2 m -1 2 ν 2 2 1 -1 z -1 m 2 ν [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ChebyshevT", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", RowBox[List["z", ",", "\[Nu]"]], "]"]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["F", "m"], "[", RowBox[List["z", ",", "\[Nu]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List["\[Nu]", "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "\[Nu]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["2", RowBox[List["1", "-", "z"]]], ")"]], "k"]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", RowBox[List["2", " ", "\[Nu]"]]]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List["-", "\[Nu]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["2", RowBox[List["1", "-", "z"]]], ")"]], "k"]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]]]], "\[Equal]", RowBox[List[RowBox[List["ChebyshevT", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["m", "+", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", RowBox[List["1", "+", "m"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["1", "+", "m"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List["\[Nu]", "-", "m", "-", "1"]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "+", "m", "-", "\[Nu]"]], ",", RowBox[List[FractionBox["3", "2"], "+", "m", "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "m"]], ",", RowBox[List["2", "+", "m", "-", RowBox[List["2", " ", "\[Nu]"]]]]]], "}"]], ",", FractionBox["2", RowBox[List["1", "-", "z"]]]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "m"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", RowBox[List["2", " ", "\[Nu]"]]]], ",", RowBox[List["1", "+", "m"]]]], "]"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["m", "-", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", RowBox[List["1", "+", "m"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List["1", "+", "m"]]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "m", "-", "1"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "+", "m", "+", "\[Nu]"]], ",", RowBox[List[FractionBox["3", "2"], "+", "m", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "m"]], ",", RowBox[List["2", "+", "m", "+", RowBox[List["2", " ", "\[Nu]"]]]]]], "}"]], ",", FractionBox["2", RowBox[List["1", "-", "z"]]]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "m"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]], ",", RowBox[List["1", "+", "m"]]]], "]"]]]]]]]]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]]]], ")"]], "&&", RowBox[List["!", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "\[Element]", "Integers"]]]]]]]]]]]]

 Date Added to functions.wolfram.com (modification date)

 2007-05-02